cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A072818 Possibly the only integers of the form sqrt(m^2*(m^2-1)*2/3) [only checked for the first 5 terms].

Original entry on oeis.org

0, 20, 1960, 192060, 18819920, 1844160100, 180708869880, 17707625088140, 1735166549767840, 170028614252160180, 16661069030161929800, 1632614736341616960220, 159979583092448300171760
Offset: 0

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Author

Henry Bottomley, Jul 14 2002

Keywords

Comments

These are the standard deviations of time for a random walk starting at 0 to reach one of the boundaries at +A001079(n) or -A001079(n) for the first time.

Examples

			0 and 20 are at the start of the sequence since m^2*(m^2-1)*2/3 (A072819) starts 0, 0, 8, 48, 160, 400, 840, 1568, ... and the only squares among these are 0 and 400 with square roots of 0 and 20.

Links

Formula

a(n) = 98*a(n-1)-a(n-2) [starting with a(0)=0 and a(1)=20] =sqrt(A072819(A001079(n))).
G.f.: 20x/(1-98x+x^2). [Philippe Deléham, Nov 18 2008]

A338429 Maximum number of copies of a 1234 permutation pattern in an alternating (or zig-zag) permutation of length n + 5.

Original entry on oeis.org

4, 8, 28, 48, 104, 160, 280, 400, 620, 840, 1204, 1568, 2128, 2688, 3504, 4320, 5460, 6600, 8140, 9680, 11704, 13728, 16328, 18928, 22204, 25480, 29540, 33600, 38560, 43520, 49504, 55488, 62628, 69768, 78204, 86640, 96520, 106400, 117880, 129360
Offset: 1

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Author

Lara Pudwell, Dec 01 2020

Keywords

Comments

The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.

Examples

			a(1) = 4. The alternating permutation of length 1+5=6 with the maximum number of copies of 1234 is 132546.  The four copies are 1246, 1256, 1346, and 1356.
a(2) = 8. The alternating permutation of length 2+5=7 with the maximum number of copies of 1234 is 1325476. The eight copies are 1246, 1256, 1247, 1257, 1346, 1356, 1347, and 1357.

Links

Crossrefs

Cf. A006325, A072819, A168380.

Formula

a(2n) = A072819(n+1) = (2*n*(n + 2)*(n + 1)^2)/3.
a(2n-1) = 4*A006325(n+1) = (2*n*(n + 1)*(n^2 + n + 1))/3.
G.f.: 4*x*(1 + x^2)/((1 - x)^5*(1 + x)^3). - Stefano Spezia, Dec 12 2021
E.g.f.: (x*(75 + 51*x + 14*x^2 + x^3)*cosh(x) + (21 + 45*x + 57*x^2 + 14*x^3 + x^4)*sinh(x))/24. - Stefano Spezia, Aug 29 2025
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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